The topological nature of the PHE is recently studied in Ref. [1]}, where
a general expression for phonon Hall conductivity is obtained in terms of the
Berry curvatures of band structures. In Ref. [1]}, the authors find a phase
transition in the PHE of the honeycomb lattice, explained from
topological nature and dispersion relations. From the Green-Kubo formula and
considering the contributions from all the phonon bands, the authors obtain
the general formula for the phonon Hall conductivity. Then by looking at
the phases of the polarization vectors of both the displacements and
conjugate momenta as a function of the wave vector, a
Berry curvature can be defined uniquely for each band. Combining the above
two steps, at last the phonon Hall conductivity can be written in terms of Berry curvatures. Such derivation gives us a clear picture of
the contribution to the phonon Hall current from all phonon branches, as well as the
relation between the phonon Hall conductivity and the geometrical phase of the
polarization vectors, which thus helps us to understand the topological picture of the PHE.
However, the process of going from the Berry phase to the heat flux and the
phonon Hall conductivity looks not very clear and natural.
